The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 3621: "Gylimic"

Scale 3621: Gylimic, Ian Ring Music Theory

Bracelet Diagram

The bracelet shows tones that are in this scale, starting from the top (12 o'clock), going clockwise in ascending semitones. The "i" icon marks imperfect tones that do not have a tone a fifth above. Dotted lines indicate axes of symmetry.

Tonnetz Diagram

Tonnetz diagrams are popular in Neo-Riemannian theory. Notes are arranged in a lattice where perfect 5th intervals are from left to right, major third are northeast, and major 6th intervals are northwest. Other directions are inverse of their opposite. This diagram helps to visualize common triads (they're triangles) and circle-of-fifth relationships (horizontal lines).

Common Names

Zeitler
Gylimic
Dozenal
Woxian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

6 (hexatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,2,5,9,10,11}

Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

6-Z40

Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

none

Reflection Axes

If a scale has an axis of reflective symmetry, then it can transform into itself by inversion. It also implies that the scale has Ridge Tones. Notably an axis of reflection can occur directly on a tone or half way between two tones.

none

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

no

Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

yes
enantiomorph: 1167

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

3 (trihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

2 (dicohemitonic)

Imperfections

An imperfection is a tone which does not have a perfect fifth above it in the scale. This value is the quantity of imperfections in this scale.

3

Modes

Modes are the rotational transformations of this scale. This number does not include the scale itself, so the number is usually one less than its cardinality; unless there are rotational symmetries then there are even fewer modes.

5

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 303

Generator

Indicates if the scale can be constructed using a generator, and an origin.

none

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[2, 3, 4, 1, 1, 1]

Interval Vector

Describes the intervallic content of the scale, read from left to right as the number of occurences of each interval size from semitone, up to six semitones.

<3, 3, 3, 2, 3, 1>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.

p3m2n3s3d3t

Distribution Spectra

Describes the specific interval sizes that exist for each generic interval size. Each generic <g> has a spectrum {n,...}. The Spectrum Width is the difference between the highest and lowest values in each spectrum.

<1> = {1,2,3,4}
<2> = {2,3,5,7}
<3> = {3,4,6,8,9}
<4> = {5,7,9,10}
<5> = {8,9,10,11}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

3.667

Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

no

Maximal Area Set

A scale is a maximal area set if a polygon described by vertices dodecimetrically placed around a circle produces the maximal interior area for scales of the same cardinality. All maximally even sets have maximal area, but not all maximal area sets are maximally even.

no

Interior Area

Area of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle, ie a circle with radius of 1.

2.116

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

5.699

Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

no

Balanced

A scale is balanced if the distribution of its tones would satisfy the "centrifuge problem", ie are placed such that it would balance on its centre point.

no

Ridge Tones

Ridge Tones are those that appear in all transpositions of a scale upon the members of that scale. Ridge Tones correspond directly with axes of reflective symmetry.

none

Propriety

Also known as Rothenberg Propriety, named after its inventor. Propriety describes whether every specific interval is uniquely mapped to a generic interval. A scale is either "Proper", "Strictly Proper", or "Improper".

Improper

Heteromorphic Profile

Defined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where c is the number of contradictions, a is the number of ambiguities, and d is the number of differences. When c is zero, the scale is Proper. When a is also zero, the scale is Strictly Proper.

(29, 16, 64)

Common Triads

These are the common triads (major, minor, augmented and diminished) that you can create from members of this scale.

* Pitches are shown with C as the root

Triad TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsF{5,9,0}131.5
A♯{10,2,5}221
Minor Triadsdm{2,5,9}221
Diminished Triads{11,2,5}131.5
Parsimonious Voice Leading Between Common Triads of Scale 3621. Created by Ian Ring ©2019 dm dm F F dm->F A# A# dm->A# A#->b°

Above is a graph showing opportunities for parsimonious voice leading between triads*. Each line connects two triads that have two common tones, while the third tone changes by one generic scale step.

Diameter3
Radius2
Self-Centeredno
Central Verticesdm, A♯
Peripheral VerticesF, b°

Modes

Modes are the rotational transformation of this scale. Scale 3621 can be rotated to make 5 other scales. The 1st mode is itself.

2nd mode:
Scale 1929
Scale 1929: Aeolycrimic, Ian Ring Music TheoryAeolycrimic
3rd mode:
Scale 753
Scale 753: Aeronimic, Ian Ring Music TheoryAeronimic
4th mode:
Scale 303
Scale 303: Golimic, Ian Ring Music TheoryGolimicThis is the prime mode
5th mode:
Scale 2199
Scale 2199: Dyptimic, Ian Ring Music TheoryDyptimic
6th mode:
Scale 3147
Scale 3147: Ryrimic, Ian Ring Music TheoryRyrimic

Prime

The prime form of this scale is Scale 303

Scale 303Scale 303: Golimic, Ian Ring Music TheoryGolimic

Complement

The hexatonic modal family [3621, 1929, 753, 303, 2199, 3147] (Forte: 6-Z40) is the complement of the hexatonic modal family [183, 1761, 1803, 2139, 2949, 3117] (Forte: 6-Z11)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3621 is 1167

Scale 1167Scale 1167: Aerodimic, Ian Ring Music TheoryAerodimic

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3621 is chiral, and its enantiomorph is scale 1167

Scale 1167Scale 1167: Aerodimic, Ian Ring Music TheoryAerodimic

Transformations:

In the abbreviation, the subscript number after "T" is the number of semitones of tranposition, "M" means the pitch class is multiplied by 5, and "I" means the result is inverted. Operation is an identical way to express the same thing; the syntax is <a,b> where each tone of the set x is transformed by the equation y = ax + b

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 3621       T0I <11,0> 1167
T1 <1,1> 3147      T1I <11,1> 2334
T2 <1,2> 2199      T2I <11,2> 573
T3 <1,3> 303      T3I <11,3> 1146
T4 <1,4> 606      T4I <11,4> 2292
T5 <1,5> 1212      T5I <11,5> 489
T6 <1,6> 2424      T6I <11,6> 978
T7 <1,7> 753      T7I <11,7> 1956
T8 <1,8> 1506      T8I <11,8> 3912
T9 <1,9> 3012      T9I <11,9> 3729
T10 <1,10> 1929      T10I <11,10> 3363
T11 <1,11> 3858      T11I <11,11> 2631
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 1671      T0MI <7,0> 3117
T1M <5,1> 3342      T1MI <7,1> 2139
T2M <5,2> 2589      T2MI <7,2> 183
T3M <5,3> 1083      T3MI <7,3> 366
T4M <5,4> 2166      T4MI <7,4> 732
T5M <5,5> 237      T5MI <7,5> 1464
T6M <5,6> 474      T6MI <7,6> 2928
T7M <5,7> 948      T7MI <7,7> 1761
T8M <5,8> 1896      T8MI <7,8> 3522
T9M <5,9> 3792      T9MI <7,9> 2949
T10M <5,10> 3489      T10MI <7,10> 1803
T11M <5,11> 2883      T11MI <7,11> 3606

The transformations that map this set to itself are: T0

Nearby Scales:

These are other scales that are similar to this one, created by adding a tone, removing a tone, or moving one note up or down a semitone.

Scale 3623Scale 3623: Aerocrian, Ian Ring Music TheoryAerocrian
Scale 3617Scale 3617: Wovian, Ian Ring Music TheoryWovian
Scale 3619Scale 3619: Thanimic, Ian Ring Music TheoryThanimic
Scale 3625Scale 3625: Podimic, Ian Ring Music TheoryPodimic
Scale 3629Scale 3629: Boptian, Ian Ring Music TheoryBoptian
Scale 3637Scale 3637: Raga Rageshri, Ian Ring Music TheoryRaga Rageshri
Scale 3589Scale 3589: Widian, Ian Ring Music TheoryWidian
Scale 3605Scale 3605: Olkian, Ian Ring Music TheoryOlkian
Scale 3653Scale 3653: Sathimic, Ian Ring Music TheorySathimic
Scale 3685Scale 3685: Kodian, Ian Ring Music TheoryKodian
Scale 3749Scale 3749: Raga Sorati, Ian Ring Music TheoryRaga Sorati
Scale 3877Scale 3877: Thanian, Ian Ring Music TheoryThanian
Scale 3109Scale 3109: Tidian, Ian Ring Music TheoryTidian
Scale 3365Scale 3365: Katolimic, Ian Ring Music TheoryKatolimic
Scale 2597Scale 2597: Raga Rasranjani, Ian Ring Music TheoryRaga Rasranjani
Scale 1573Scale 1573: Raga Guhamanohari, Ian Ring Music TheoryRaga Guhamanohari

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages are ©2005 William Zeitler (http://allthescales.org) used with permission.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (An algorithm for spelling the pitches of any musical scale) Contact authors Patent owner: Dokuz Eylül University, Used with Permission. Contact TTO

Tons of background resources contributed to the production of this summary; for a list of these peruse this Bibliography. Special thanks to Richard Repp for helping with technical accuracy, and George Howlett for assistance with the Carnatic ragas.